Simplify and expand the following expression: $ \dfrac{5}{a + 4}- \dfrac{1}{a - 5}- \dfrac{5a}{a^2 - a - 20} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{5a}{a^2 - a - 20} = \dfrac{5a}{(a + 4)(a - 5)}$ Now we have: $ \dfrac{5}{a + 4}- \dfrac{1}{a - 5}- \dfrac{5a}{(a + 4)(a - 5)} $ The least common multiple of the denominators is: $ (a + 4)(a - 5)$ In order to get the first term over $(a + 4)(a - 5)$ , multiply by $\dfrac{a - 5}{a - 5}$ $ \dfrac{5}{a + 4} \times \dfrac{a - 5}{a - 5} = \dfrac{5(a - 5)}{(a + 4)(a - 5)} $ In order to get the second term over $(a + 4)(a - 5)$ , multiply by $\dfrac{a + 4}{a + 4}$ $ \dfrac{1}{a - 5} \times \dfrac{a + 4}{a + 4} = \dfrac{a + 4}{(a + 4)(a - 5)} $ Now we have: $ \dfrac{5(a - 5)}{(a + 4)(a - 5)} - \dfrac{a + 4}{(a + 4)(a - 5)} - \dfrac{5a}{(a + 4)(a - 5)} $ $ = \dfrac{ 5(a - 5) - (a + 4) - 5a} {(a + 4)(a - 5)} $ Expand: $ = \dfrac{5a - 25 - a - 4 - 5a}{a^2 - a - 20} $ $ = \dfrac{-a - 29}{a^2 - a - 20}$